The Standards for Mathematical Practice, Part 1

I have been looking to write a series unpacking the Standards for Mathematical Practice (SMP) since reading a great text about Common Core Math & PLC’s edited by Timothy Kanold (see Resources); however, other topics have pulled me away from that goal.  Recently though, Heinemann has charged some of their “mathematics heavyweights” with writing a series on the Standards, releasing one article for each SMP.  So, with that as inspiration, I have been sharing & synthesizing the articles for my staff.

See below to read about Standards 1-4, and stay tuned for Part 2 in my series that will cover the Standards 5-8.  Enjoy, and feel free to share some of your own bite-sized best practices in the comments section below!

SMP #1: Make Sense of Problems & Persevering in Solving Them – Problem solving is one of the hallmarks of mathematics (Kinold, et al.), and Sue O’Connell encourages us to consider the process (sequencing), strategies & attitude of our students as we teach them to solve problems.  In planning for instruction we must provide good problems, which Kinold, et al. describe as having the characteristics listed below.

• Relevant to Students
• Meaningful Mathematics to Advance Understanding
• Opportunities to Apply/Extend Skills
• Support Multiple Strategies

Before we are ready to tackle the problem, our main focus must be on comprehension (Yes, I am putting my “literacy hat” back on).  While we typically focus on key words (i.e. how many left), consider the role that close reading routines can play as students work.  The CUBES framework, is one simple way to move beyond “key words” and can help reveal student understanding.

Kanold, et al. also provide a great framework to support students as they take “responsibility of organizing their thoughts about tackling the problem.”

SMP #2: Reason Abstractly & Quantitatively – In her blog entry, Pam Harris encourages us to promote quantitative reasoning by providing opportunities for students to contextualize & decontextualize numbers. Two important benefits to this reasoning is that students will be able to apply concepts in meaningful ways & (more importantly) reconstruct faded knowledge by reasoning with the content (Kanold, et al.). According to Kanold, et al., opportunities for students to develop number sense should include activities where students can:

• Express Interpretations about Numbers: Working with equal to/less than/greater than number sets, placed in context (i.e. pounds, length, etc.)
• Apply Relationships Between Numbers: Using the same numbers but in different contexts (as presented in Harris’ article)
• Recognize Magnitude of Numbers: Is 20 a big number? When eating cheeseburgers it is; when counting stars it is not. The discourse that accompanies this between teacher-student & student-student is essential!
• Compute: How many ways can you make the number 42?
• Make Decisions Involving Numbers: Is 15 minutes enough time to…? Where is the 5th house on the left?
• Solve Problems: Constructing/Deconstructing numeric operations culminates with problem solving activities. This is where explicit modeling should definitely take place & can be the focus of our mini-lessons. This can even be utilized as a “close reading” activity!

SMP #3: Constructing Viable Arguments & Critique the Reasoning of Others – Steve Leinwand begins by stating that this standard contains the “nine most important words in the entire Common Core” and I couldn’t agree more! In addressing this standard, our students must move beyond simply solving the problem, to focusing on how the problem is solved. As Leinwand states, they must be encouraged to communicate, justify, critique & reflect. This will ensure that our focus shifts from a single approach, to “many approaches & justifications, thereby strengthening everyone’s learning.” In short, our classroom environment must provide opportunities for students to:

• Provide explanations & justifications as part of their solution processes (making & evaluating conjectures)
• Make sense of their classmates’ solutions by asking questions for clarification
• Facilitate meaningful discussions about mathematics

Click here to read my entry on “sentence stems” that help to support purposeful talk and here to see my take on student questioning & dialog (through the lens of Danielson’s FFT).  Although our focus here is on mathematics, the scaffolding tools provided can surely be utilized across the curriculum.

SMP #4: Model with Mathematics – One of the things that is particularly interesting about SMP #4 (Model with Mathematics) is how explicitly it links to SMP #3 & SMP #5. In his article, Dan Meyer explains 5 essential pieces to SMP #4, which are outlined below.

• Identify Essential Variables: As we plan, it is essential that we provide REAL WORLD situations as a focus for student work (Meyer uses the register line, for example). This will help students to extract the necessary information/facts that will allow them to solve the problem (for the literacy folks, this is “determining importance”). 
• Formulate Models: When formulating models, we are talking about “nonlinguistic representations.” However, Kanold et al., reminds us that we must avoid focusing solely on manipulatives, as nonlinguistic representation also consists of diagrams, charts, graphs…and NUMBERS! See below for a nice planning framework that was included with the Kanold text.

• Performing Operations: Here is where students will apply their math skills by synthesizing the essential variables & models to determine a conclusion.
• Interpreting Results & Validating Conclusions: Do the conclusions make sense…why/why not?

Most importantly, Meyer reminds us that “modeling goes wrong when we do the most interesting parts for our students.” It is essential that we avoid “leaving them to focus solely on performing operations & interpreting results.” Instead, we must provide students the opportunity to “[make predictions], speculate which information matters, and decide what matters most.”

Click on the graphic below to visit Heinemann’s site & view each of the SMP articles that influenced this piece.